Introductory Notes on Public Goods
for Intermediate Microeconomics
Let’s begin with an extremely simple example of a public good. Suppose there are only
two people who live on the shore of Lake Magnavista. Amy likes to water ski and Bev likes
to sunbathe. Both activities are seriously affected by the level of the water in the lake. When
there is a lot of water in the lake, it’s good for water skiing but the water line is so high that
there is no beach for sunbathing. When there is much less water, the sunbathing is good
but the lake is too shallow for water skiing. Therefore Amy prefers that the lake have lots
of water, and Bev prefers that it have much less water. Fortunately, it’s possible to raise or
lower the water level costlessly, by opening a dam at one end of the lake or at the other end.
Unfortunately, it’s not clear at what level the water ought to be set.
In order to have a measure of the amount of water in the lake, let’s use the water’s depth
at a specified location on the lake: let x denote the water’s depth (in feet) at that location.
Amy’s and Bev’s preferences are described by the following utility functions:
uA (x, yA ) = yA − (15 − x)2
and
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uB (x, yB ) = yB − (6 − x)2 ,
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where x denotes the water level and yA and yB are Amy’s and Bev’s daily consumption of
other goods, measured in dollars. Suppose Amy and Bev each have incomes of $100 per day.
Note that Amy’s and Bev’s marginal rates of substitution are
M RSA = 30 − 2x
and
M RSB = 6 − x.
Amy’s most-preferred water level is x
bA = 15 and Bev’s most-preferred level is x
bB = 6. Figure
1 depicts their indifference maps. Notice that if the water level is above x
bB , Bev would be
willing to pay (i.e., to give up some of the y-good) to have x reduced, and that Amy would
similarly be willing to pay to reduce x if it’s above her ideal level, x
bA .
What makes this situation different than everything we’ve seen before is that the x variable
can’t be at different levels for different people. It’s not like pizza or beer, where Amy can
consume one quantity and Bev a different quantity. In this case, the water level can be varied,
but it will be the same for both of them. That’s why we haven’t used A and B subscripts on
the x variable: it’s just one variable, not two. The water level in this example is a public
good.
Figure 1
Let’s try to determine which outcomes are Pareto efficient. Let’s start by asking whether a
water level of x = 8 feet is efficient. Figure 2 will be helpful here: it’s a diagram similar to
the Edgeworth Box. The difference is that here in this diagram the “corners” or “origins”
for both consumers are placed on the left edge of the box, so that when x increases (i.e., as
we move toward the right), both persons’ consumption of the x-good is increasing. This is in
contrast to the Edgeworth Box, where as we move toward the right, Person A’s consumption
is increasing and B’s is decreasing, because with “private goods” like pizza or beer, the more
one person gets, the less is available for the other person.
Figure 2
At x = 8, Amy’s and Bev’s MRS’s will be M RSA = 14 and M RSB = −2. Amy would be
willing to pay $14 to increase the water level one foot and Bev would be willing to pay $2 to
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decrease it by one foot — or Bev would be willing to accept a payment of $2 as compensation
for increasing the water level by one foot. Therefore, if we were to increase the water level
by a foot, and if Amy were to compensate Bev by paying her, say, $6, then they would both
be better off. In fact, you can calculate that Amy’s utility will increase from 51 to 58 and
that Bev’s utility will increase from 98 to 101 21 .
So what about this new water level of x = 9 feet — is it efficient? We have M RSA = 12
and M RSB = −3, so we could increase the water level by another foot, with Amy paying
another $6 to compensate Bev. The $6 payment is less than the $12 Amy would have been
willing to pay, and more than the $3 Bev would have been willing to accept as compensation,
so they’re again both better off from the one-foot increase with $6 compensation. You can
calculate that their utilities will have increased again, to uA = 63 and uB = 104.
Now it’s becoming clear that as long as Amy would be willing to pay more for an increase
than Bev would be willing to accept as compensation, then such a bargain — increasing
the water level, with Amy compensating Bev — will make them both better off. In other
words, the water level is not Pareto efficient so long as M RSA > −M RSB — i.e., so long as
M RSA + M RSB > 0. When M RSA + M RSB > 0 the marginal social value of an increase
in x is positive, so x should be increased. Similarly, we could show that if M RSA +M RSB < 0,
then x should be decreased (because the marginal social value of an increase is negative, so
the marginal social value of a decrease in x is positive).
The Pareto efficient outcomes are therefore the ones that satisfy the Test Condition M RSA +
M RSB = 0. In our example, it’s easy to solve for the efficient water level in the lake:
M RSA + M RSB = (30 − 2x) + (6 − x) = 36 − 3x;
therefore the Test Condition yields
36 − 3x = 0
i.e.,
x = 12.
The efficient water level is 12 feet, where M RSA = 6 and M RSB = −6.
Now we know the water level that’s Pareto efficient. But what level will actually be chosen?
As usual, that depends on the institutional arrangements that are used for choosing the water
level. For example, the affected parties might vote on the level they want. The outcome of
that institution can be analyzed using game theory, which we won’t do here. Let’s suppose
instead that Bev owns the lake, or at least that she has the right to choose the water level.
What will the water level be? It seems that Bev will choose the level that she likes best,
namely x = 6 feet.
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But we’ve already seen that at such a low water level, Bev would be better off to allow
a higher level if Amy will compensate her appropriately. Indeed, just exactly as in our
discussion earlier in the semester of bargaining between two parties, we would expect them
to arrive at a mutually agreeable bargain in which there are no more gains to be had from
further trade or bargaining — i.e., at a Pareto efficient outcome. In our example this means
that the water level will be 12 feet, with Amy paying Bev some amount as compensation for
the increase from 6 to 12 feet. In Figure 3 they will end up on the vertical line at x = 12, with
Amy paying Bev an amount that leaves their y consumptions between their initial indifference
curves, the ones that pass through the allocation in which x = 6 and yA = yB = 100.
Figure 3
What if instead Amy has the right to choose the water level? At first it seems as if she would
choose the level she likes best, x = 15 feet. But the same argument as in the preceding
paragraph tells us that again we should expect the two women to bargain with one another
to set the water level at x = 12 feet, but this time with Bev paying some compensation to
Amy for setting the water level below what Amy would like the most.
It was the economist Ronald Coase who pointed out that the assignment of legal rights will
affect only who pays compensation, and how much, but will not affect the level of the activity
that affects both parties. He was awarded the Nobel Prize for this idea, which has had a
profound effect on law and, to a lesser extent, legislation in recent decades.
What if the public good is not costless, as the water level was? Then the Pareto Efficiency
Test Condition (PETC) is that the marginal social value of a one-unit increase in the amount
of the public good must be equal to the marginal cost of producing that additional unit —
i.e., that the sum of all the MRS’s must be equal to MC:
(P ET C)
M RS1 + M RS2 + · · · + M RSn = M C.
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Notice that in our lake example the MC of changing the water level was zero, so this Test
Condition is actually the same as the one we used in the example, just generalized to account
for costly public goods as well as costless ones.
Another example: Suppose Amy and Bev are plagued by mosquitoes, but it’s possible to
control the number of mosquitoes by spraying regularly. However, the mosquito spray can’t
be confined to the property of just one of the women: any spray that’s applied affects the
entire lake shore equally. Suppose Amy’s and Bev’s preferences are described by the same
utility functions as in our water-level example, where x now denotes the number of gallons
that are sprayed each week, and yA and yB still denote the amounts Amy and Bev spend on
other goods. If each gallon of spray were free, our problem would be exactly the same as
before, because we would have M C = 0. But suppose instead that the spray costs $24 per
gallon. Then our Test Condition (PETC) yields the following:
M RSA + M RSB = M C :
(30 − 2x) + (6 − x) = 24;
i.e., 36 − 3x = 24.
Therefore the Pareto efficient amount of mosquito spray is 4 gallons per week, which equates
the marginal social value to the marginal social cost, $24.
How much will be sprayed if Amy and Bev each choose on the basis of just their own personal
benefits from the spray? Bev won’t choose to spray at all, because even when x = 0 her MRS
is much less than the price of the spray ($6 vs. $24). Amy will choose x = 3 gallons of spray,
which equates her own MRS to the $24 price of the spray. So the amount of spray that’s
chosen will be less than the Pareto efficient amount. Both women would be better off if an
additional gallon were sprayed and Amy were to pay, say, $22 of the cost of the additional
gallon and Bev paid the remaining $2.
A more striking example: Suppose there are 100 homeowners living on the lake shore,
and that each one has M RS = 6 − x, where x is the number of tankfuls that are sprayed.
Suppose that the cost of the mosquito spray is $100 per tankful. Then the Pareto efficient
amount of spray is x = 5 tankfuls:
X
M RSi = M C : 100(6 − x) = 100;
i.e., 600 − 100x = 100.
How much will actually be sprayed if each homeowner chooses on the basis of just his own
benefit from the spray? At x = 0 each homeowner has M RS = 6 — i.e., each would be
willing to pay $6 to increase x from x = 0 to x = 1 tankful sprayed. But the cost of such an
increase — $100 — far exceeds each person’s MRS. Therefore no one will choose to spray.
Even though the marginal social benefit is $600 and the marginal cost is only $100, no spray
is forthcoming.
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